Paper · Dec 6, 2021

Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain

Jiwoong Jang

math.APWider flowsarXiv:2112.02715v3Calculus of Variations and Partial Differential Equations, 62, 2023
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Abstract

In this paper, we study the forced mean curvature flows and the prescribed mean curvature equations of both graphs and level-sets with capillary-type boundary conditions on a bounded domain, which is not necessarily convex. We prove a priori gradient estimates locally Lipschitz in time. Under an assumption on the forcing term, we prove that the gradient estimates are globally Lipschitz in time. As a consequence, we obtain the existence theorem of solutions. In our formulation, we recover the known results of the gradient estimates on a strictly convex bounded domain. Next, we study the associated eigenvalue problems for mean curvature flows of both graphs and level-sets. We prove the large time behavior of the solutions of mean curvature flows of graphs on a smooth bounded domain. Finally, we compute the asymptotic speed of the solutions of level-set mean curvature flows and the large time profile of level-sets in the radially symmetric case based on optimal control formula. Examples arising in the radially symmetric case demonstrate that the additional assumption on the forcing term is optimal.

Comments: 59 pages

Mean curvature flowVolume-preserving & constrained flowsBoundaries & non-compactFree boundary & capillaryRotational symmetry & warped productsEigenvalues & spectrum

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)1Jang 2021 A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting (Jiwoong Jang, 2023)5Jang 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)3Mitake 2023 A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain (Tokuhiro Eto and Yoshikazu Giga, 2024)8Eto 2024 Diffusion-aggregation equations and volume-preserving mean curvature flows (Jiwoong Jang and Antoine Mellet, 2025)6Jang 2025 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)7Jang 2025 Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions (Linlin Fan and Peibiao Zhao, 2026)4Fan 2026 Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain (Tokuhiro Eto and Jiwoong Jang, 2026)2Eto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Boundaries & non-compact · Same author Jiwoong Jang · Shared terms “large time”, “level set”, “forced mean”

  2. Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

    Tokuhiro Eto and Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Same author Jiwoong Jang · Shared terms “bounded domain”, “smooth bounded”, “level set”

  3. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

    Hiroyoshi Mitake and Hung V. TranWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Boundaries & non-compact · Shared terms “large time”, “level set”, “forced mean”

  4. Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

    Linlin Fan and Peibiao ZhaoWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Boundaries & non-compact · Shared terms “capillary type”, “type boundary”, “boundary condition”

  5. A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows · Same author Jiwoong Jang · Shared terms “forced mean”

  6. Diffusion-aggregation equations and volume-preserving mean curvature flows

    Jiwoong Jang and Antoine MelletWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same author Jiwoong Jang · Shared terms “forcing term”, “formulation”

  7. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same author Jiwoong Jang

  8. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “bounded domain”, “level set”, “set mean”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.